** SPECIAL
NOTE RE MD-DC-VA Germantown Meeting Presentation:
**

**I made an error in my presentation at the Germantown meeting, 11/5/05.
The envelope or boundary curve for the region swept out by sliding
a ladder around a corner may not arise as the discriminant of a quadratic
equation, as I claimed. My error was in confusing two different families
of lines. For the ladder problem, when I attempt to ascertain whether
or not a given point lies on a line in the family of lines, I end up with
a quartic equation. It may be that a different algebraic approach might
lead to a quadratic, but this seems dubious to me now. It may also
be that the discriminant of the quartic vanishes on the envelope or boundary
curve, but I have not verified that.
**

**The quadratic comments are accurate in the context of a different family
of lines, one for which the sum of the x and y intercepts is
a fixed constant. This is actually a common design for string art.
For example, take one line with y intercept 9 and x intercept
1, then create a second line with y intercept 8 and x
intercept 2, and continue in like fashion, repeatedly incrementing the x
and y intercepts by 1 and -1, respectively. This is shown in
the figure below.**

If we let

**(10 - p) x + py = p(10 - p)**

**The envelope in this case is a parabola with axis of symmetry along
the line y = x, and the equation of the parabola can be obtained by
using the discriminant of a quadratic equation.
**

**Here is how that works: let ( x , y) be a fixed
but arbitrary point in the first quadrant, and ask Does this point lie
on the line for some value of the parameter p? To find out, solve
the equation for p in terms of x and y. The equation
is clearly quadratic in p and can be expressed in the form**

**and the discriminant is given by D = (y - x -
10)**

**( y - x - 10)**

**for this family of lines.**